Area-under-the-curve Formula Sheet

The area under the curve measures the total space between a graph of a function and the x-axis, giving a visual representation of the integral of that function.

Neetesh Kumar | June 04, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1.Curve Tracing
• 2.Area Under the Curves

1. Curve Tracing:

The following outline procedure is to be applied in sketching the graph of a function $(y = f(x))$, which in turn will be extremely useful to quickly and correctly evaluate the area under the curves.

(a) Symmetry: The symmetry of the curve is judged as follows:

• If all the powers of $(y)$ in the equation are even, then the curve is symmetrical about the $(x)$ axis.

• If all the powers of $(x)$ are even, the curve is symmetrical about the $(y)$ axis.

• If powers of $(x)$ & $(y)$ both are even, the curve is symmetrical about the axis of $(x)$ as well as $(y)$.

• If the equation of the curve remains unchanged on interchanging $(x)$ and $(y)$, then the curve is symmetrical about $(y=x)$.

• If on replacing 'x' by '-x' and 'y' by '-y', the curve's equation is unaltered, there is symmetry in opposite quadrants, i.e., symmetric about the origin.

(b) Find $( \frac{dy}{dx})$ and equate it to zero to find the points on the curve where you have horizontal tangents.

(c) Find the points where the curve crosses the x-axis & also the y-axis.

(d) Examine if possible the intervals when $( f(x))$ is increasing or decreasing. Examine what happens to ‘y’ when $(x \to \infty)$ or $( x \to -\infty)$.

2. Area Under the Curves:

(a) Whole area of the ellipse, $( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1)$ is $( \pi ab)$.

(b) Area enclosed between the parabolas $(y^2 = 4ax)$ & $(x^2=4by)$ is $(\frac{16ab}{3})$.

(c) Area included between the parabola $(y^2=4ax)$ & the line $(y=mx)$ is $(\frac{8a^2}{3m^3})$.

(d) Area enclosed by the parabola and its double ordinate $(P,Q)$ is two-thirds of the area of rectangle $(PQRS)$, where $(R,S)$ lie on the tangent at the vertex.

(e) The area bounded by the curve $y=f(x)$, the x-axis, and the ordinates $x=a$ & $x=b$ is given by:
$A =\int_{a}^{b} f(x)\,dx =\int_{a}^{b}y\,dx,\quad f(x) \ge 0$

(f) If the area is below the x-axis then $(A)$ is negative. The convention is to consider the magnitude only, i.e.,
$A = \int_{a}^{b} y \, dx$

in this case.

(g) The area bounded by the curve $x=f(y)$, y-axis & abscissa $y=c$ , $y=d$ is given by:
$\text{Area} =\int_{c}^{d} f(y)\,dy =\int_{c}^{d} x\,dy,\quad f(y)>0$

(h) Area between the curves $y =f(x)$ & $y = g(x)$ between the ordinates $x=a$ & $x=b$ is given by:
$A =\int_{a}^{b}f(x)\,dx -\int_{a}^{b}g(x)\,dx =\int_{a}^{b}f(x)-g(x)\,dx ,\quad f(x)>g(x)\;\forall x\in(a, b)$

(i) Average value of a function $(y=f(x))$ with respect to $(x)$ over an interval $(a \leq x \leq b )$ is defined as:
$y_{\text{av}} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx$

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